True False[1 Marks ]

Question 11 Mark

State whether statement are True or False.
If the sum of $n$ terms of a sequence is quadratic expression, then it always represents an A.P.

Answer

False.
Solution:
We know that the sum of n terms of A.P. is
$\text{S}_\text{n}=\frac{\text{n}}{2}(2\text{a}+(\text{n}-1)\text{d})=\frac{\text{n}}{2}(2\text{a}-\text{d}+\text{nd})$
$=\Big(\frac{2\text{a}-\text{d}}{2}\Big)\text{n}+\Big(\frac{\text{d}}{2}\Big)\text{n}^2$
Thus, $S_n $ is of type $An^2 + Bn.$
But general quadratic expression is of the form $An^2 + Bn + C.$
Thus, if the sum of $n$ terms of a sequence is quadratic expression of type $An^2 + Bn + C,$
where $\text{C}\neq0,$ it does not represents sum of A.P.

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Question 21 Mark

State whether statement are True or False.
Any term of an A.P. (except first) is equal to half the sum of terms which are equidistant from it.

Answer

True.
Solution:
Let a be the first term and d be the common difference of the A.P.
Consider any term ar of an A.P.
Now, $a_{r+m} = ar + (m - 1)d$
And $a_{r-m} = ar + (m - 1)(-d)$
$\therefore a_{r+m} + a_{r-m}= a_r + (m - 1)d + a_r + (m - 1)(-d)$
$\Rightarrow a_{r+m} + a_{r-m} =2a_r$
$\Rightarrow\text{a}_\text{r}=\frac{\text{a}_{\text{r}+\text{m}}+\text{a}_{\text{r}-\text{m}}}{2}$
Thus, any term of an A.P. (except first) is equal to half the sum of terms which are equidistant from it.

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Question 31 Mark

State whether statement are True or False.
The sum or difference of two G.P.s, is again a G.P.

Answer

False.
Solution:
Let two G.P.s are $\text{a},\text{ar}_1,\text{ar}_1^2,\text{ar}_1^3....;$ and $\text{b},\text{br}_2,\text{br}_2^2,\text{br}_2^3,....$
Now, sum of two G.P.s is $\text{a}+\text{b},(\text{ar}_1+\text{br}_2),\big(\text{ar}_1^2+\text{br}_2^2\big),....$
Clearly,
$\frac{\text{ar}_1+\text{br}_2}{\text{a + b}}\neq\frac{\text{ar}_1^2+\text{br}_2^2}{\text{ar}_1+\text{br}_2}$
Similarly, for difference of two G.P.s, we get
$\frac{\text{ar}_1+\text{br}_2}{\text{a + b}}\neq\frac{\text{ar}_1^2+\text{br}_2^2}{\text{ar}_1+\text{br}_2}$
So, the sum or difference of two G.P.s is not a G.P.

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Question 41 Mark

State whether statement are True or False.
Two sequences cannot be in both A.P. and G.P. together.

Answer

True.Solution:
Let us consider G.P, a, ar and $ar^2$
If it is in A.P then $\text{ar}-\text{a}\neq\text{ar}^2-\text{ar}$
Hence, the given statement is True.

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Question 51 Mark

State whether statement are True or False.
Every progression is a sequence but the converse i.e., every sequence is also a progression need not necessarily be true.

Answer

True.Solution:
Let us consider a sequence of prime number 2, 3, 5, 7, 11, ….
It is clear that this progression is a sequence but sequence is not a progression because it does not follow a specific pattern. Here, the given statement is True.

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